The rocket fuel needed to produce that 40 MJ weighs close to 1 kg, especially when you include the oxidiser. So the energy needed to accelerate 1kg of payload to LEO velocity is much more.

That whole "tyranny of the rocket equation" thing is why I am surprised the actual first stage for launching a rocket is NOT a ground based reusable "up-chucker".

Basically, I would have thought that any momentum that can be imparted to the rocket before it has to rely on its self propulsion would be a huge help. Not talking about eliminating self propulsion, just an assist so the rocket could carry a larger payload or be smaller or whatever.

IE like a variation on Jules Verne's big gun for throwing the payload up there but engineered to be plausible and having the rocket still be self propelled. And safe.

But we don't seem to do this. So why?

Edit: First part of video [0]. Apparently it's not completely dumb. Just stupid-hard/impossible to do practically at the size required for big rockets and payloads. But small ones might work. Maybe.

[0] https://www.youtube.com/watch?v=lWYn5hl4QWg

Building your rocket to survive the upchucker costs more than the savings from being upchucked.

Chuckers are the optimal large scale solution for airless bodies, but they're horizontal. You spread the acceleration out over a very long distance so you don't need a super beefy spacecraft and your humans won't turn to goo. Basically, a maglev train except it has track above as well as below and it doesn't have a maximum speed. Wrap one around the lunar equator and it can eject anywhere from sundiver to interstellar escape with human-tolerable acceleration.

Except that you save maybe 30% of the cost to just launch from Earth. Once youre off planet you're over half way to anywhere, and you don't need to land on the Moon to go further

The structural support needed to keep the rocket from crumpling under the throw is extra weight to carry the rest of the way.

This is why Starship is aiming for a catch rather than legs. Legs that work at 1g added too much weight.

I've also wondered about a balloon launch. Strap the rocket to an enormous blimp, it handles the initial 1-2m/s acceleration, saving an enormous first volley of rocket fuel needed to break objects at rest out of their state of rest.

After the rocket is clear, activate compressors inside of the blimp and return it to base for re-use.

But the energy needed is not an indicator of what is difficult or dangerous. Leaving the atmosphere intact is the most difficult part of launching a rocket going by failure rate. Of those that reach space, those that still fail often took damage from the launch.

Once you're in space, force over distance until your fuel runs out.

Failures rate is even less a suitable indicator. Going up 100km is achievable by a simple single stage solid fuel rocket. Going to orbit requires way way more complexity, including a giant first stage that can fail in atmosphere.